Can I say "What are you doing today?" when I want to say "Hi!" to a person?
As expressed in previous answer, it is too literal to use as a greeting.
You could perhaps use "What's up?" instead to try to achieve a similar thing.
This is treated as a greeting whilst literally meaning "what are you doing?" / "how are you?".
The receiver can then either just say a greeting back or answer either question above.
No, those phrases are not the same. "Hi" is a greeting, while "What are you doing today?" is used when you want to know what that person's schedule will be like that day. Why would you want to know what they are doing today? Maybe you want to spend time with them (if they are busy then you can't), or you are simply curious as to what they will be doing later that same day.
However, there is a similar phrase to "what are you doing today?" It is "how are you doing today?"
Other person: "Hello!"
You: "Hi! How are you (doing) today?"
Other person: "Great, how are you?"
You: "Great, thanks."
I put the "doing" in parentheses because the sentence would still make sense if it was not there.